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2009-06-11, 08:32   #342
davieddy

"Lucan"
Dec 2006
England

2×3×13×83 Posts

Quote:
 Originally Posted by Prime95 and a second prime95 cannot serve as the "official" double-check because it doesn't pass the "independent software" requirement.
Pity, because Kevin (with 4 cores) was due to complete at the same
time as Tony.
Can you shed any light on the interim residue discrepancy, or would
that let the cat out of the bag?

2009-06-11, 08:58   #343
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

18C316 Posts

Quote:
 Originally Posted by davieddy Can you shed any light on the interim residue discrepancy
Well it would be easy to test for yourself. Just use a very small number like p=31 or something and print residues every iteration for first and double check modes. You can then compare to a manually computed set of residues. Easy.

Why all the guessing? It can be proved in a matter of minutes!

2009-06-11, 09:02   #344
akruppa

"Nancy"
Aug 2002
Alexandria

2,467 Posts

Quote:
 Originally Posted by Mr. P-1 Can "slightly better" be quantified? Could this be tested empirically by seeing if those p where p-1 is smooth really do have fewer factors?
I have no idea how to quantify this. An empirical test is the best I can think of. Only relatively small divisors should be affected, so one might check if those 2[I]p[/I] where p-1 has at least n divisors are more likely to survive trial division to, say, 240.

Alex

2009-06-11, 09:51   #345
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

27738 Posts

Quote:
 Originally Posted by philmoore I didn't consider the probabilities of more than 8, which does raise the figures somewhat. However, we can't consider the 40 possible sequences as independent, so I would guess that 12.9% is an overstatement of the true probability, but I have no idea how much.
I think it is better to do simulation, using http://primes.utm.edu/notes/faq/NextMersenne.html conjecture for the probability that Mp is prime. 2000 simulations for the [2,5*10^7] interval, gives:
Code:
The most number of Mersenne primes in a [x,2.06*x] interval (for the exponent),
what we have after the verification for M40-M47 are 8 primes.
3 Mersenne primes in 'small' interval: 5
4 Mersenne primes in 'small' interval: 110
5 Mersenne primes in 'small' interval: 536
6 Mersenne primes in 'small' interval: 710
7 Mersenne primes in 'small' interval: 432
8 Mersenne primes in 'small' interval: 162
9 Mersenne primes in 'small' interval: 40
10 Mersenne primes in 'small' interval: 5
So it isn't very rare 8 or more primes, the probability is 10.35%, close to the easy Poission estimation.
I've also counted the number of Mersenne primes in the runs. Note that using the conjecture the expected number of primes is about 48.9 (I've included p=2 in every cases).
Code:
26 Mersenne primes: 1
27 Mersenne primes: 0
28 Mersenne primes: 2
29 Mersenne primes: 0
30 Mersenne primes: 1
31 Mersenne primes: 4
32 Mersenne primes: 2
33 Mersenne primes: 3
34 Mersenne primes: 4
35 Mersenne primes: 13
36 Mersenne primes: 15
37 Mersenne primes: 13
38 Mersenne primes: 39
39 Mersenne primes: 37
40 Mersenne primes: 41
41 Mersenne primes: 66
42 Mersenne primes: 72
43 Mersenne primes: 81
44 Mersenne primes: 94
45 Mersenne primes: 100
46 Mersenne primes: 117
47 Mersenne primes: 95
48 Mersenne primes: 134
49 Mersenne primes: 121
50 Mersenne primes: 113
51 Mersenne primes: 134
52 Mersenne primes: 130
53 Mersenne primes: 106
54 Mersenne primes: 93
55 Mersenne primes: 86
56 Mersenne primes: 62
57 Mersenne primes: 56
58 Mersenne primes: 36
59 Mersenne primes: 36
60 Mersenne primes: 27
61 Mersenne primes: 23
62 Mersenne primes: 11
63 Mersenne primes: 10
64 Mersenne primes: 9
65 Mersenne primes: 3
66 Mersenne primes: 5
67 Mersenne primes: 2
68 Mersenne primes: 1
69 Mersenne primes: 1
70 Mersenne primes: 0
71 Mersenne primes: 0
72 Mersenne primes: 0
73 Mersenne primes: 1

Last fiddled with by R. Gerbicz on 2009-06-11 at 09:52

2009-06-11, 10:45   #346
davieddy

"Lucan"
Dec 2006
England

2×3×13×83 Posts

Quote:
 Originally Posted by R. Gerbicz I think it is better to do simulation, using http://primes.utm.edu/notes/faq/NextMersenne.html conjecture for the probability that Mp is prime. 2000 simulations for the [2,5*10^7] interval, gives: Code: The most number of Mersenne primes in a [x,2.06*x] interval (for the exponent), what we have after the verification for M40-M47 are 8 primes. 3 Mersenne primes in 'small' interval: 5 4 Mersenne primes in 'small' interval: 110 5 Mersenne primes in 'small' interval: 536 6 Mersenne primes in 'small' interval: 710 7 Mersenne primes in 'small' interval: 432 8 Mersenne primes in 'small' interval: 162 9 Mersenne primes in 'small' interval: 40 10 Mersenne primes in 'small' interval: 5 So it isn't very rare 8 or more primes, the probability is 10.35%, close to the easy Poission estimation.
Since you have chosen x to 2.06x as your interval,
and 2.06 is M47/M40 (exponents of) I don't think we should
count both M40 and M47 in that interval. So we should call it
a run of 7 rather than 8 primes, much more likely (32%)..

Last fiddled with by davieddy on 2009-06-11 at 10:51

2009-06-11, 11:05   #347
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

1,531 Posts

Quote:
 Originally Posted by davieddy Since you have chosen x to 2.06x as your interval, and 2.06 is M47/M40 (exponents of) I don't think we should count both M40 and M47 in that interval. So we should call it a run of 7 rather than 8 primes, much more likely (32%)..
For x=20996011, after the verification we'll know 8 primes in [x,2.06*x],
2.06*x>43112609.

Last fiddled with by R. Gerbicz on 2009-06-11 at 11:06

2009-06-11, 11:57   #348
davieddy

"Lucan"
Dec 2006
England

2×3×13×83 Posts

Quote:
 Originally Posted by R. Gerbicz For x=20996011, after the verification we'll know 8 primes in [x,2.06*x], 2.06*x>43112609.
Not unless x is chosen carefully i.e. very close to M40.
This is bias.
I still think you are "double counting" like saying the are 8 primes in
the interval M1-M8, 8 in the interval M8-M16...etc.

Another way of looking at it is to ask for the probability of finding
8 or more primes AFTER M39 all <= 43112609, giving us anterval of [x,3*x]
(roughly)

PS I might not have understood exactly what you did in your simulation.
Did you generate mock sequences of "M Primes" based on the probability, then
take x to be each prime in turn, then count the number of primes
>=x and < 2.06*x ?

.

Last fiddled with by davieddy on 2009-06-11 at 12:33

2009-06-11, 12:56   #349
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

1,531 Posts

Quote:
 Originally Posted by davieddy Not unless x is chosen carefully i.e. very close to M40. This is bias. I still think you are "double counting" like saying the are 8 primes in the interval M1-M8, 8 in the interval M8-M16...etc. Another way of looking at it is to ask for the probability of finding 8 or more primes AFTER M39 all <= 43112609, giving us anterval of [x,3*x] (roughly) PS I might not have understood exactly what you did in your simulation. Did you generate mock sequences of "M Primes" based on the probability, then take x to be each prime in turn, then count the number of primes >=x and < 2.06*x ? .
OK, I understand you, choosing r as ratio of two exponents of Mersenne primes increase the count in an "unfair" way. What I wanted to point out that even for r=2.06 the 8 Mesenne primes isn't a big surprise, a 10% event isn't very rare.

Yes, the simulation was that: by probab(p)=if(p%4==3,a=2,a=6);return(1.781*log(a*p)/(p*log(2))) probability I've choosen Mp as prime for every p>2. Note that the interseting fact that probab(3) and probab(5)
are higher than 1 so M2,M3,M5 were always in the sequence as primes.

The another way would be: simulate also the sequence of primes: if n>1 then by 1/log(n) probability choose n as prime and if it is choosen prime then by probab(p) probability Mp is prime. However this simulation is slower than the previous and I think in theory it gives the same distribution.

2009-06-11, 13:37   #350
davieddy

"Lucan"
Dec 2006
England

2×3×13×83 Posts

Quote:
 Originally Posted by R. Gerbicz What I wanted to point out that even for r=2.06 the 8 Mesenne primes isn't a big surprise, a 10% event isn't very rare.
That is exactly what I meant by "No need to invoke divine intervention"
in my post which initiated this discussion.

2009-06-11, 14:43   #351
Kevin

Aug 2002
Ann Arbor, MI

433 Posts

Quote:
 Originally Posted by davieddy Pity, because Kevin (with 4 cores) was due to complete at the same time as Tony. Can you shed any light on the interim residue discrepancy, or would that let the cat out of the bag?
Apparently there wasn't supposed to be any interim residue discrepancy (besides the +2 shift between glucas/mlucas and mprime/prime95, which was accounted for). No point in finishing the test if the interim residues show that something already went wrong.

2009-06-11, 14:46   #352
mdettweiler
A Sunny Moo

Aug 2007
USA (GMT-5)

624910 Posts

Quote:
 Originally Posted by Kevin Apparently there wasn't supposed to be any interim residue discrepancy (besides the +2 shift between glucas/mlucas and mprime/prime95, which was accounted for). No point in finishing the test if the interim residues show that something already went wrong.
Are you sure, though, that you weren't simply testing the wrong exponent from the start? If that's the case, then the interim residuals would be different no matter what, and you may as well finish the test rather than letting the already completed work go to waste. No harm in doing a plain old doublecheck on an exponent that needs it.

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